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Full text of "Treatise On Analysis Vol-Ii"


Let (Ea)ael be any family of uniformizable spaces, and let E = fj Ea.


For each ael, let (dat^)^eL<x ^e a family of pseudo-distances defining the
topology of Ea. If we put

(12.5.1)                          e^(x,y) = da^(prxx9 ptay)

for each pair (x, y) of elements of E = ]^[ Ea, then it is immediately checked


that the ea> A are pseudo-distances on E. The product of the topologies of the
Ea is then the topology on E defined by the ea^ as a runs through I and, for
each a e I, A runs through La. The set E endowed with this topology is called
the product of the spaces Ea. When I = {1,2} this definition agrees with that
given above for two spaces E1? E2.

For every finite family (a, Ai)1^i^n and every finite family (r^igign f
strictly positive real numbers, the set B(x; ((a, A)), (rf)) can be written in
the form f| Ba where, if a = af, we have Ba = B(praix; (A,-), (r,-)), where j


runs through the indices for which a,- = a, and Ba = Ea if a is not equal to
any oct. From this and (3.6.4) we deduce immediately:

(12.5.2)    For each ael, the projection pra is a continuous mapping 0/E onto Ea,
and the image under pra of any open set in E is an open set in Ea.

(12.5.3)    For each finite subset H of I, and each family (Ua)aeH, -where Ua is
open in Ea/0r each a e H, the set I fj Ua\ x / fj   Ea\ is open in E.

\aeH       /         \aeI-H     /

For it is the intersection of the sets pr^UJ, where a e H, and these sets
are open (12.5.2) and finite in number.

The sets described in (12.5.3) are called elementary sets in E. The descrip-
tion given above of the neighborhoods shows that (having regard to (3.9.3))
these sets form a basis for the topology of E. Incidentally, this shows also that
the topology of E depends only on the topologies of the Ea and not on the
families of pseudo-distances (rfa> A)- Moreover, if Sa is a basis for the topology
of Ea, the elementary sets such that Ua e 23a for all a form a basis for the
topology of E. It should be noted that, if I is infinite, a product f] Ua with


Ua open in Ea, nonempty and ^ Ea for all ael, is not open in E.

(12.5.4)    Let Aa be a subset of Ea, for each a, and let A = f] Aa. Then


A = | Aa. In particular, if each Aa is closed in Ea, then A is closed in E.

aelo-distances on E, and the definition of neigh-